I've been in Adelaide for the last three years and have had as much work as I've wanted - but all short contracts. I haven't tried for a permanent position or a longer contract but I think it's possible if you have experience.

University work tends to be more abundant in the 10 weeks before a new semester starts but can then drop off quite quickly.

I'm not sure about the ELICOS centres or the migrant work but I think it's quite healthy.

I'm teaching high school ESL in Adelaide, but I've got a teaching degree. When I came back from China 2 years ago, I taught a 10 week contract at LM Training Specialists doing migrant ed, and then started a 14 week ELICOS one at Flinders Uni when the permanent high school job came up.
The other unis also run ELICOS, TAFE does migrant ed, SACE runs international student classes. So there is a market, though they don't advertise much. A direct approach maybe better.
As noted, they tend to employ in short term contracts, though it apppears that most get renewed. Migrant Ed classes run for 40 weeks of the year, but you only get vacation pay for 2 or 3, so do the sums!